Selecting the lorenz parameters for wideband radar
waveform generation
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Citation
Willsey, Matt S., Kevin M. Cuomo, and Alan V. Oppenheim.
“Selecting the lorenz parameters for wideband radar waveform
generation.” International Journal of Bifurcation and Chaos 21.09
(2011): 2539–2545.
As Published
http://dx.doi.org/10.1142/s0218127411029914
Publisher
World Scientific
Version
Author's final manuscript
Accessed
Thu May 26 08:49:44 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/72687
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
1
Selecting the Lorenz Parameters for Wideband
Radar Waveform Generation
Matt S. Willsey+ , Kevin M. Cuomo† , and Alan V. Oppenheim‡
Abstract
Radar waveforms based on chaotic systems have occasionally been suggested for a variety of radar applications.
In this paper, radar waveforms are constructed with solutions from a particular chaotic system, the Lorenz system,
and are called Lorenz waveforms. Waveform properties, which include the peak autocorrelation function side-lobe
and the transmit power level, are related to the system parameters of the Lorenz system. Additionally, scaling the
system parameters is shown to correspond to an approximate time and amplitude scaling of Lorenz waveforms and
also corresponds to scaling the waveform bandwidth. The Lorenz waveforms can be generated with arbitrary time
lengths and bandwidths and each waveform can be represented with only a few system parameters. Furthermore, these
waveforms can then be systematically improved to yield constant-envelope output waveforms with low autocorrelation
function sidelobes and limited spectral leakage.
Index Terms
Radar, Waveforms, Wideband, Chaos, Lorenz, Time-Scaling.
I. I NTRODUCTION
Due to the aperiodic nature of chaotic dynamics, chaotic systems can be used to generate signals with low
autocorrelation sidelobes. Since many radar systems utilize the matched filter to achieve high range resolution and
maximize the SNR in Gaussian noise [North, 1963], low autocorrelation function sidelobes help prevent false alarms
and prevent stronger returns from masking the presence of weaker returns. Furthermore, for wide bandwidths, these
chaotic waveforms lead to high range resolution [Rihaczek, 1996].
Many waveform design techniques have been proposed using chaotic systems with amplitude or frequency/phase
modulation (see, for example, [Venkatasubramanian & Leung, 2005; Lukin, 2001; Flores, et al., 2003]). The
contribution of this paper is to develop a specific chaotic system, the Lorenz system [Lorenz, 1963], for waveform
∗
This work is sponsored in part under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, recommendations and conclusions
are those of the authors and are not necessarily endorsed by the United States Government.
+
Matt S. Willsey was with the Research Laboratory of Electronics while the research for this manuscript was conducted.
†
Kevin M. Cuomo (cuomo@ll.mit.edu) is with M.I.T. Lincoln Laboratory.
‡
Alan V. Oppenheim (avo@mit.edu) is with Research Laboratory of Electronics and is supported in RLE in part by BAE Systems PO
112991, Lincoln Laboratory PO 7000032361, and the Texas Instruments Leadership University Program.
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generation by appropriately selecting the system parameters. Specifically, scaling the Lorenz parameters is shown
in Section II to approximately scale the Lorenz solutions in time and amplitude, which can therefore be used to set
the bandwidth of the Lorenz waveforms. The time-scaling property also allows a decoupling of the three system
parameters into one parameter that controls the bandwidth and two parameters that control other waveform properties.
The two-dimensional parameter space is explored in Section III to determine how these parameters affect various
waveform properties. Whereas techniques with amplitude modulation often sacrifice sensitivity and techniques with
frequency/phase modulation often lead to spectral leakage (due to abrupt transitions in instantaneous frequency),
the resulting Lorenz waveforms can be improved using the procedure in [Willsey et al., 2010] to yield high-power
waveforms with limited spectral leakage. Moreover, since each waveform is uniquely determined by a combination
of the system parameters and initial conditions, each waveform can be stored with six system parameters.
II. T IME S CALING S OLUTIONS OF THE L ORENZ E QUATIONS
The Lorenz system, as given in Eq. 1, is one of the most well-studied chaotic systems [Strogatz, 1994; Cuomo,
1994; Guckenheimer & Holmes, 1983].
ẋ =
σ(y − x)
ẏ
=
rx − y − xz
ż
=
xy − bz
(1)
In order for the Lorenz system to give rise to chaotic dynamics, the Lorenz parameters σ, r, and b must satisfy
Eqs. 2 - 4.
b >
0
(2)
σ
>
b+1
(3)
r
>
σ(σ + b + 3)
≡ rc .
(σ − b − 1)
(4)
The variable rc in the above equation is the critical value for r.
In this section, two techniques useful in generating time-scaled solutions to the Lorenz equations are presented.
The first technique modifies the Lorenz equations so that the resulting solutions exactly equal time-scaled solutions
of the original system. The second technique gives rise to approximately time-scaled solutions by scaling the Lorenz
parameters.
To simplify the representation of the Lorenz equations, we use the vector x, as defined in Eq. 5, to denote the
x, y, and z state variables and use the vector f(x), as defined

x


x =  y

z
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in Eq. 6, to denote the Lorenz equations.



.

(5)
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

σ(y − x)
f (x) =


 rx − y − xz

xy − bz


.

(6)
With this notation, the Lorenz system in Eq. 1 is expressed as
ẋ = f (x).
(7)
A. Exactly Time-Scaling the Lorenz Equations
Using the above notation, a slightly modified Lorenz system is presented in Eq. 8 where a > 0. As can be seen
in Eq. 8, the original Lorenz equations are scaled by a. The solutions, x1 (t), of Eq. 8 equal time-scaled solutions
to the Lorenz equations, x(at), for identical initial conditions1 , i.e.
x˙1
=
af (x1 )
x1 (t) =
(8)
x(at).
(9)
The fact that the system in Eq. 8 gives rise to time-scaled solutions of the Lorenz equations, Eq. 9 can be verified
by differentiating x1 (t) and using the chain rule to obtain
x˙1 (t)
Substituting f (at) for
d[x(at)]
d[at]
=
d[x(at)]
a
d[at]
and using Eq. 9 results in
x˙1
=
af (x1 (t))
(10)
The time dependence can be dropped for notational convenience to yield the differential equations in Eq. 8, which
shows that these equations are the differential equations yielding the solutions in Eq. 9.
The time-domain relationship in Eq. 9 also implies a frequency-domain relationship between the Fourier transform
of x(t) and x1 (t). Specifically, a time scaling by a results in a frequency scaling by
1
a.
Correspondingly, the
bandwidth of x1 (t) increases by a factor of a as the value of a in Eq. 8 increases. This bandwidth scaling is a
useful property when generating radar waveforms.
B. Approximate Time-Scaling of the Lorenz Equations through the Lorenz Parameters
In this section, appropriate selection of the Lorenz parameters is shown to result in an approximate time and
amplitude scaling of the Lorenz solutions. A system that corresponds exactly to a time and amplitude scaling is
given in Eq. 11. As expressed in Eq. 12, solutions to Eq. 11, x2 (t), are equal to time and amplitude scaled solutions
1 Even
though this relationship is exact, due to rounding in the numerical integration methods and to the divergence of nearby Lorenz
trajectories, numerically demonstrating the relationship in Eq. 9 is difficult if the magnitude of a is too large.
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to the Lorenz equations if the initial conditions are related as in Eq. 13. These equations can be verified as was
done previously with Eqs. 8-9.


aσ(y2 − x2 )
ẋ2

1

= a2 f ( x2 ) =  arx2 − ay2 − x2 z2
a

x2 y2 − ab2 z2
x2 (t) =




(11)
(12)
ax(at)
{x2 (0), y2 (0), z2 (0)} = {ax(0), ay(0), az(0)}
(13)
Equating Eq. 11 to a Lorenz system of the form of Eq. 1 with parameters σ̂, r̂, b̂ would require that




σ̂
aσ








 r̂  =  ar − x̂ŷ (a − 1)  .




b̂
ab
(14)
Consequently, a choice for a constant r̂ does not exist, and the Lorenz parameters cannot be chosen to exactly time
and amplitude scale the corresponding solutions.
However, when the term x̂ŷ (a − 1) in Eq. 14 is negligible for most values of t, then selecting σ̂ = aσ, r̂ = ar, and
b̂ = ab results in solutions x̂(t) where x̂(t) ≈ x2 (t) = ax(at). In [Willsey, 2006], the term
ŷ
x̂ (a − 1)
is determined
to be negligible when
r
≥
360
a
∈
[0.4, ∞)
(15)
Summarizing, we choose
{σ̂, r̂, b̂} = {aσ, ar, ab}
(16a)
{x̂(0), ŷ(0), ẑ(0)} = {ax(0), ay(0), az(0)}
(16b)
with
and the constraints in Eq. 15, in which case
x̂(t) ≈ ax(at)
(17)
To illustrate time-scaling through the use of a parameter scaling, two Lorenz systems with scaled parameters were
numerically integrated (using a 4th order Runge-Kutta method with a step size of 10−3 seconds) and compared.
The first system has a set of parameters {σ, r, b} equal to {267, 595, 100}. A set of initial conditions were found
on the attractor of this Lorenz system and were used to generate the x state variable which is shown with a solid
line in Fig. 1. The parameters and the initial conditions of the second system were chosen according to Eq. 16 with
a = 0.95. The x state variable of the second system is plotted with a dashed line in Fig. 1. As expected, the signal
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σ = 267, r = 595, b = 100
600
σ = 254, r = 565, b = 95
Amplitdue
400
200
0
−200
−400
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (s)
Fig. 1.
Time Evolution of the x State-Variable for Two Lorenz Systems. The two systems have different parameter sets that give rise to two
Lorenz systems where one system is a time-scaled and re-normalized version of the other system. The time-scaling and re-normalization factor
was 0.95.
represented with the dashed line is time and amplitude scaled by a factor of 0.95 when compared to the signal
represented with the solid line2 . Although only x(t) is shown, y(t) and z(t) are similarly scaled3 .
III. S ELECTING THE L ORENZ S YSTEM PARAMETERS FOR WAVEFORM G ENERATION
A typical radar waveform s(t) is a real signal of the form shown in Eq. 18 (with a(t) and θ(t) slowly varying
relative to ωc ), and the spectral content of these signals is concentrated around ±ωc [Rihaczek, 1996].
s(t) =
a(t) cos(ωc t + θ(t))
(18)
The corresponding complex envelope, µ(t), and the transmit waveform, s(t) are related as shown in Eq. 194 .
s(t) =
<{µ(t)ejωc t }
(19)
All of the waveforms presented in this paper are used as the complex envelope.
The radar waveforms are extracted from the Lorenz system as a normalized, time-windowed segment of the x
state variable5 . The Lorenz waveform, xL (t), is expressed mathematically in Eqs. 20 and 21 where T denotes the
2 Since
the time and amplitude scaling is approximate, the two trajectories will eventually diverge as a result of a chaotic system’s sensitivity
to initial conditions.
3 It
is worth noting that although an exact time scaling via the Lorenz parameters is not attainable, the Lorenz system can be changed to give
other chaotic systems with solutions that can be exactly time scaled by scaling its system parameters. Such systems are proposed in [Willsey,
2006].
4 For
a more detailed discussion of this analytic formulation of transmit waveform, see Chapter 2 of [Rihaczek, 1996].
5 Based
on preliminary studies, x(t) was shown to lend itself best to waveform design when compared to the other state variables or
combinations of state variables.
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length of the waveform in time and xp denotes the maximum of |x(t)| over the time interval t ∈ [0, T ].
xL (t)
=
wr (t)
=
1
wr (t)x(t)
xp

 1; t ∈ [0, T ]
 0; else
(20)
(21)
These waveform can also be generated with arbitrary time lengths and bandwidths. An additional benefit to these
Lorenz waveforms is that, due to the uniqueness of solutions, each Lorenz waveform is uniquely specified by the
initial conditions, the time length, and the Lorenz parameters, which allows for convenient compression and storage
of the entire waveform that was generated in terms of a very small number of parameters. The remaining focus of
this section is to select the Lorenz parameters so that the waveforms generated from the Lorenz system have the
desired radar waveform properties.
A. Decoupling the Lorenz Parameters
As shown in Sec. II, scaling the system parameters approximately scales in time and amplitude the Lorenz
solutions, and therefore also scales the bandwidth. For most parameter values, the Lorenz parameters can be
decoupled such that one parameter primarily determines the bandwidth and the other two parameters control other
radar waveform properties. This decoupling simplifies the determination of the desired waveform properties.
To decouple the Lorenz parameters so that one parameter controls bandwidth and the other two are used to
control other waveform properties, we introduce the scaling parameters a, K and express σ, r, and b in Eq. 1 as
σ
=
aσ0
r
=
ar0
b =
aK
(22)
By first selecting K in b = aK, all combinations of {σ, r, b} can be reached by varying σ0 , r0 , and a. When
the constraints in Eq. 15 hold, varying a is interpreted as scaling the bandwidth by a, as discussed in Sec. II. As
discussed below, the relevant combinations of system parameters for waveform generation satisfy Eq. 15.
In the next two sections, the Lorenz parameter space is explored. For this discussion the value of K is chosen to
be 100, and the values of σ0 and r0 in Eq. 23 are varied to improve the average transmit power and range sidelobes
of the radar waveform.
{σ, r, b}
= a · {σ0 , r0 , 100}
(23)
After determining σ0 and r0 , a can be varied to give the desired system bandwidth.
B. Average Transmit Power
Many practical radar systems have peak power limitations. Consequently, transmit waveforms are usually designed
with an average transmit power within a few dB of peak power. The peak-to-RMS ratio (PRMS) of the Lorenz
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1.6
2.9
1.5
2.8
ro /rc
1.4
2.7
2.6
1.3
2.5
1.2
2.4
2.3
1.1
2.2
1
200
Fig. 2.
300
400
500
σo
600
700
800
Averaged Peak-to-RMS Ratio of Lorenz waveforms for Combinations of σ0 and r0 . The variable rc is the critical value for r.
waveforms, which are used as the complex envelope µ(t), quantifies this loss from peak transmit power. A low
value for PRMS is desired.
The Lorenz parameters can be varied to reduce the PRMS ratio of Lorenz waveforms. With a = 1 in Eq. 23, the
PRMS is explored for σ0 ∈ (200, 850) and r0 ∈ (rc , 1.6rc )6 (where rc is given in Eq. 4). Using each combination
of σ0 and r0 , three distinct, 100 second Lorenz waveforms are generated with randomly selected initial conditions.
The PRMS value associated with that combination of parameters is then determined by averaging the PRMS value
over three trials. The results are shown in Fig. 2. As shown in Fig. 2, the PRMS is lowest in the lower, left-hand
corner when both σ0 and r0 are reduced. In [Willsey, 2006], this PRMS variation is shown to be influenced by the
underlying Lorenz dynamics also dependent on the Lorenz parameters.
C. Range Sidelobes
Since many practical radar systems utilize matched filters, the autocorrelation function, as defined in Eq. 24 is a
critical design consideration7 :
Z
∞
rµµ (t) =
µ(τ )µ∗ (τ − t)dτ.
(24)
−∞
Since the time scale of the matched-filter response is proportional to range separation between the target and
the radar, autocorrelation function sidelobes correspond to sidelobes in range. Thus, low autocorrelation-function
sidelobes prevent false alarms and also prevent larger targets from interfering with smaller targets. For Lorenz
waveforms, the magnitude of the peak side-lobe of the autocorrelation function can be reduced by appropriately
selecting the system parameters.
6 Preliminary
7 To
empirical studies indicated that these ranges were most relevant to waveform design.
simplify this discussion, the Doppler effect has been ignored.
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1.6
dB:
1.5
−5
ro /rc
1.4
−10
1.3
−15
1.2
−20
1.1
−25
1
200
Fig. 3.
300
400
500
σo
600
700
800
Peak Side-lobe of the Lorenz System for Various Combinations of σ0 and r0 in Eq. 23 with a = 1.
The relationship between the peak side-lobe and the Lorenz parameters, as defined in Eq. 23, is determined
empirically for various combinations of σ0 ∈ (200, 850) and r0 ∈ (rc , 1.6rc )8 while a = 1. For each combination
of σ0 and r0 , the peak autocorrelation function side-lobe is determined by averaging the peak side-lobe of three
distinct Lorenz waveform with randomly selected initial conditions. The results are shown in Fig. 3, which illustrates
that a region exists that corresponds to a minimum peak side-lobe.
The connection between the peak autocorrelation function side-lobe and the Lorenz parameters is further illustrated
in Fig. 4. In this plot, the autocorrelation function corresponding to two different sets of system parameters are
compared. The first set of system parameters, {σ = 262, r = rc , b = 100}, lies in the darkest-shaded, low sidelobe region of Fig. 3; whereas, the second set of parameters, {σ = 262, r = 1.6 · rc , b = 100}, lies in a region
corresponding to a relatively high peak side-lobe. As expected the peak side-lobe associated with the first set of
system parameters is lower than the peak side-lobe associated with the second set of parameters. The dependence
of the peak side-lobe on the Lorenz parameters results from underlying Lorenz system properties as discussed in
[Willsey, 2006]9 .
D. Design Curves for the Lorenz Parameters
By using Fig. 2 and Fig. 3, the values of σ0 and r0 in Eq. 23 can be chosen to trade off the peak-to-RMS
ratio with the peak autocorrelation function side-lobe. This trade-off is illustrated with the design curves for xL (t)
8 Preliminary
9 In
studies indicated that these ranges were most relevant to waveform design.
summary, as a trajectory moves around one wing of the Lorenz attractor, there is a probability, p, at which the trajectory transitions
to the other wing of the attractor and a probability (p − 1) at which the trajectory remains on the same wing of the attractor. The value of
p is empirically shown to depend on the Lorenz parameters, and the Lorenz parameters corresponding to p = 0.5 are shown to also be the
parameters that correspond to the minimum peak side-lobe.
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0
L L
¯
¯
(t) ¯
¯r
20 log10 ¯ rxxLxxL (0) ¯ (dB)
−10
−20
−30
−40
−50
−60
−70
−80
−0.15
−.1
−0.05
0
0.05
0.1
0.15
Time Delay (s)
Fig. 4.
Comparison of Two Normalized Autocorrelation Functions, rxL xL , from Two Systems with Different System Parameters. The solid
blue curve illustrates the autocorrelation function for the Lorenz parameters, {σ = 262, r = rc , b = 100}, that correspond to a minimal peak
autocorrelation function side-lobe. The dashed red curve illustrates the autocorrelation function for the Lorenz parameters, {σ = 262, r =
1.6 · rc , b = 100}, that correspond to a relatively high peak side-lobe.
shown in Fig. 5. The solid line, C1 , represents the combinations of σ0 and r0 that give rise to Lorenz waveforms
with autocorrelation functions possessing minimized peak sidelobes. The three dashed lines represent combinations
of σ0 and r0 that give rise to Lorenz waveforms with a peak-to-RMS ratio roughly equal to 2.2, 2.3, and 2.4,
respectively. As depicted in Fig. 2, the peak-to-RMS ratio of the Lorenz waveform monotonically increases from
the bottom left corner of Fig. 5 to the top right corner. When generating xL (t), combinations of σ0 and r0 will be
chosen on C1 below the line corresponding to a peak-to-RMS less than 2.3. These combinations of parameters are
illustrated in the figure as the operating region, which is the shaded portion of C1 10 . Finally, the parameter a in
Eq. 23 can be varied to set the bandwidth of the resulting Lorenz waveform.
IV. D ISCUSSION ON THE R ESULTING L ORENZ WAVEFORMS
The Lorenz waveforms described in this paper possess many properties that make them well-suited for several
applications. With large bandwidths and long time lengths, waveforms from the Lorenz system possess sharp
autocorrelation functions capable of achieving high range resolution and low range sidelobes. Furthermore, waveform
generation with the Lorenz system is useful in generating a large set of nearly orthogonal waveforms (as described
in [Willsey et al., 2010]) for applications such as those relevant to MIMO radar systems [Robey et al., 2004]. As
described in Sec. III, these waveforms can be generated with arbitrary time lengths and bandwidths, and each of
these waveforms can be represented with only a few, finite-precision system parameters. Finally, since scaling the
10 All
lines in Fig. 5 are drawn with a relatively large width to indicate that the combinations of σ0 and r0 that give rise to minimized
sidelobes and particular realizations of the peak-to-RMS ratio were numerically approximated (as opposed to analytically determined).
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1.6
PRM
2.4
2.3
= 2.2
S=
S=
PRM
PRMS
1.5
1.4
C
ro /rc
1
1.3
Operating
Region
1.2
Minimum
Side−lobes
1.1
1
200
250
300
σo
350
400
450
Fig. 5. Lorenz Waveform Design Curves. For b0 = 100, these curves indicate combinations of σ0 and r0 that tradeoff the peak-to-RMS ratio
and magnitude of the peak side-lobe.
Lorenz parameters is shown to approximately time scale the Lorenz solutions, the potential exists for obtaining
range and Doppler characteristics of a target by using scaled Lorenz parameters either with the self-synchronizing
property in [Pecora & Carroll, 1991] or with a method similar to that presented in [Liu et al., 2007] with Chua’s
circuit.
Although the Lorenz parameters have been determined to achieve desirable radar waveform properties, an
evaluation of these waveforms in [Willsey et al., 2010] suggests that they can be improved even further with
the procedure in [Willsey et al., 2010]. This improvement results in radar waveforms based on the Lorenz system
that possess low range sidelobes, an average power within a few dB of peak power, and limited spectral leakage.
Plots of the corresponding autocorrelation function and energy spectrum are depicted in [Willsey et al., 2010].
Figure 6 from [Willsey, 2006] depicts a typical transmit signal to illustrate how the average power of the waveform
is within a few dB of peak power.
V. C ONCLUSIONS
In this paper, the relationship between the Lorenz parameters and the radar waveform properties of the Lorenz
waveforms is explored. Selection of the Lorenz parameters is shown to vary the Lorenz waveform bandwidth - via a
time and amplitude scaling - and is shown to trade-off the PRMS ratio with peak autocorrelation function side-lobe.
Due to the naturally tapered spectrum and bounded dynamics of the Lorenz solutions, the resulting waveforms can
be readily improved with the procedure in [Willsey et al., 2010].
ACKNOWLEDGMENT
This work was supported in part by under Air Force Contract FA8721-05-C- 0002. Opinions, interpretations,
recommendations and conclusions are those of the authors and are not necessarily endorsed by the United States
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1.5
1.25
1
0.75
Amplitude
0.5
0.25
0
−0.25
−0.5
−0.75
−1
−1.25
−1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (µs)
Fig. 6.
A Time Segment of a Radar Waveform Based on the Lorenz System. This waveform was originally generated as a Lorenz waveform
and was improved with an improvement procedure in [Willsey et al., 2010] and [Willsey, 2006].
Government. This work was supported in RLE in part by BAE Systems PO 112991, Lincoln Laboratory PO
7000032361, and the Texas Instruments Leadership University Program. The authors would like to thank the Siebel
Foundation for the funding from the Siebel Scholarship Program. A special thanks goes to Dr. Frank Robey at
M.I.T. Lincoln Laboratory for his help and support. Additionally, the contributions from all the members of the
Digital Signal Processing Group at M.I.T. and Group 33 at M.I.T. Lincoln Laboratory were also appreciated.
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Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139.
Available at: http://www.rle.mit.edu/dspg/documents/mattnewthesis.pdf
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Robey, F. C., Coutts, S., Weikle, D., McHarg, J. C. & Cuomo, K. M. [2004], ”MIMO radar theory and experimental results,” in Signals, Systems
and Computers, 2004, Conference Record of the Thirty-Eighth Asilomar Conference, Pacific Grove, CA.
Pecora, L. M. & Carroll, T. L. [1991] ”Synchronization in chaotic systems,” Physical Review Letters, 64, 821-824.
Liu, Z., Zhu, X., Hu, W. & Jiang, F. [2007] ”Principles of Chaotic Signal Radar,” International Journal of Bifurcation and Chaos, 17, 1735-1739.
Matt S. Willsey Matt S. Willsey attended the Massachusetts Institute of Technology where he received his B.S. degree in electrical science and
engineering in 2005 and his M.Eng. degree in electrical engineering and computer science in 2007.
Mr. Willsey was a graduate student at the Research Laboratory of Electronics at M.I.T. during this project, and his research interests included
waveform design, nonlinear dynamics, and MIMO radar systems. He is currently employed as an Engineering Scientist at the Applied Research
Laboratories, the University of Texas at Austin (ARL:UT). At ARL:UT, his his research is focused on signal processing applications for sonar
systems. He was the recipient of the Siebel Scholarship in 2005 and is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi.
Kevin M. Cuomo Kevin M. Cuomo received the B.S. (magna cum laude) and M.S. degrees in electrical engineering from the State University
of New York (SUNY) at Buffalo, in 1984 and 1986, respectively, and the Ph.D. degree in electrical engineering from Massachusetts Institute
of Technology (MIT), Cambridge, in 1994.
He is a Senior Staff Member in the Ranges and Test Bed Group, MIT Lincoln Laboratory, where he works on the development of advanced
signal processing methods for various radar applications. Before joining Lincoln Laboratory in 1988, he was a member of the research staff at
Calspan Corporation, Buffalo, NY. He has authored several technical papers and reports in the areas of super resolution data processing and
applications of chaotic dynamical systems for radar and communications. Recent research efforts include the development of super orthogonal
waveforms for Multiple-Input Multiple-Output (MIMO) radar and corresponding beam space signal processing techniques.
Dr. Cuomo was awarded Departmental Honors and was the recipient of a University Fellowship and a Hughes Fellowship while attending
SUNY Buffalo. He is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi.
Alan V. Oppenheim Alan V. Oppenheim was born in New York, New York on November 11, 1937. He received S.B. and S.M. degrees in
1961 and an Sc.D. degree in 1964, all in Electrical Engineering, from the Massachusetts Institute of Technology. He is also the recipient of an
honorary doctorate from Tel Aviv University.
In 1964, Dr. Oppenheim joined the faculty at MIT, where he is currently Ford Professor of Engineering. Since 1967 he has been affiliated
with MIT Lincoln Laboratory and since 1977 with the Woods Hole Oceanographic Institution. His research interests are in the general area of
signal processing and its applications. He is coauthor of the widely used textbooks Discrete-Time Signal Processing and Signals and Systems.
He is also editor of several advanced books on signal processing.
Dr. Oppenheim is a member of the National Academy of Engineering, a fellow of the IEEE, and a member of Sigma Xi and Eta Kappa Nu.
He has been a Guggenheim Fellow and a Sackler Fellow. He has received a number of awards for outstanding research and teaching, including
the IEEE Education Medal, the IEEE Jack S. Kilby Signal Processing Medal, the IEEE Centennial Award and the IEEE Third Millennium
Medal. From the IEEE Signal Processing Society he has been honored with the Education Award, the Society Award, the Technical Achievement
Award and the Senior Award. He has also received a number of awards at MIT for excellence in teaching, including the Bose Award and the
Everett Moore Baker Award.
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